Root mean squared radius of WS

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The Woods-Saxon potential is

f(r) = 1/(1+Exp(\frac{r-R}{a}))

where R=r_0 A^{1/3} is the half-maximum radius that r_0 is the reduced radius and A is the nuclear mass number, and a is the diffusiveness parameter.

The root-mean-squared (rms) radius is

\sqrt{\left<r^2\right>} = \sqrt{\int{f(r) r^2 r^2 dr} / \int{f(r) r^2 dr}}

where the “extract” r^2 is because of spherical coordinate.

The integration is a polynomial

\int{f(r) r^n dr} = -a^{n+1} \Gamma(n+1) \sum_{k=1}^{\infty} \frac{(-Exp(R/a))^k}{k^{n+1}}

In mathematica, the sum is notated by,

PolyLog(n, x) = \sum_{k=1}^{\infty} \frac{x^k}{k^n}

Thus, the rms radius for a > 0 is

\sqrt{\left<r^2\right>} = a \sqrt{12 \frac{PolyLog(5, -Exp(R/a))}{PolyLog(3, -Exp(R/a))} }

For a = 0 , f(r) = 1, r < R,

\sqrt{\left<r^2\right>} = \sqrt{\frac{3}{5}}R

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Woods-Saxon Shape

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this is a collective model of the nuclei density vs radius. it has another name Fermi-shape.

\rho(r) = \frac { \rho_0} {1+ Exp( (r - R_0)/a) }

where \rho_0 is central density, or density at r = 0. R_o is the radius of half density and a is the diffuseness. when a is large, the “tail” of the shape will be longer.

the radius is measured in unit of fm 1 fm = 10^{-15} m .

R_0 can be vary from 1 fm to 7 or 8 fm.  for 16O, it is about 2 fm. and for 208Pb, it is about 6fm .

a is more or less the same for different nuclei.

the density of nuclei can be mass density or charge density, the Woods-Saxon also gives a good approximation.

for mass density, a 10^{17} kg m^{-3} . for comparison, water density is 1 kg per meter cube.

and charge density is about 0.15 e fm^{-3} .